#232884

order	512
nilpotency class	3
isoclinism	isoclinic class of #232876
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,15)(5,8)(6,9)(7,13)(11,12)(14,16)(17,18)(19,29)(20,33)(21,26)(22,27)(23,28)(24,31)(25,34)(30,38)(32,39)(35,43)(36,42)(37,40)(41,47)(44,46)(45,48);(1,3)(2,7)(4,14)(5,13)(6,12)(8,11)(9,16)(10,15)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,41)(26,29)(27,39)(28,40)(33,44)(34,46)(35,45)(36,47)(38,43)(42,48);(1,2)(3,11)(4,8)(5,15)(6,9)(7,13)(10,12)(14,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,42)(27,33)(28,43)(29,44)(31,45)(32,47)(37,46)(39,48)(40,41);(3,12)(7,16)(10,11)(13,14)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,34,36,35)(24,39,40,29)(25,42,43,33)(30,45,47,46)(38,48,41,44)

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140080);